Integral

An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals along with the derivatives are the fundamental objects of calculus. Other words include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in Physics and elementary calculus. An integral is a number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or area measurement of that unit and all such products are summed. The independent variables may be confined within certain limits (definite integral) or in the absence of limits (indefinite integral).

Every definition of an integral is based on a particular measure. For instance, Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The process of computing an integral is called integration (a more archaic term for integration is quadrature) and the appropriate computation of an integral is termed numerical integration. There are two classes of Riemann integrals: definite integrals which have upper and lower limits and indefinite integrals.

 Terminology and Notation

If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the intergrand. The region over which a function is being integrated is called the domain of integration. Usually this domain will be an interval, in which case it is enough to give the limits of that interval, which are called the limits of integration. In general, the integrand may be a function of more than one variable and the domain of integration may be an area, volume, a higher dimensional region or even an abstract space that does not have a geometric structure in any usual sense.

Formal definitions

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. One of the most common integral is the Lebesgue integral.

Lebesgue Integral

 The Lebesgue integral achieves greater flexibility by directing attention to the weight in the weighted sum. Hence the definition begins with a measure. However this integral is not used for a wide range of functions and applications, as there seems to be lot of loop holes in the theorem.

Questions:

• What is an Integral?
• What are intergrand, domain of integration and the limits of integration?